26-Jun-10
PHYS102 – 12
CAPACITORS IN SERIES AND PARALLEL
Purpose
1.
2.
To measure the capacitance of a capacitor.
To investigate the capacitance of capacitors in series and in parallel.
Introduction
The performance of many circuits can be predicted by systematically combining various
circuit elements in series or parallel into their equivalents.
For capacitors the equivalent capacitance for series and parallel combinations is as follows:
1/Cs = 1/C1+1/C2 + ....+ 1/Cn
C1
≡
C2
Cs
For two capacitors
1/Cs = 1/C1 + 1/C2
Cn
or
Cs = C1C2/(C1 + C2).
(1)
SERIES
----------------------------------------------------------------------------------------------------------------------
Cp = C1 + C2 + .... + Cn
C1
C2
Cn
≡
Cp
For two capacitors
Cp = C1 + C2 .
PARALLEL
© KFUPM – PHYSICS
revised 26/06/2010
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Department of Physics
Dhahran 31261
(2)
26-Jun-10
PHYS102 – 12
Method
If a capacitor is charged to a certain voltage, and then disconnected from the voltage source,
the voltage on the capacitor will stay at the same value for a long time (determined by the
leakage resistance of the capacitor).
Vo
Vo
Vo
Vo
….retains
voltage Vo
Capacitor charged
to Vo
C
R
C
C
0
….discharges
However, if the capacitance is connected to a resistance R, it will discharge; the time it takes
to discharge is governed by R and C. Circuit theory indicates that the voltage at time t after
the voltage source is disconnected is:
V = Vo e-(t/RC)
(3)
Here Vo is the initial voltage and e is the base of natural or Naperian logarithms, e =
2.71828...
In this experiment we will measure the "1/k" time, T1/k, that is, the time needed for the
voltage to change from Vo to Vo/k. Here k is a number in the range 1 < k < ∞ .
You are perhaps familiar with certain T1/k values:
"half-life"
T1/2 = 0.693 RC
(k = 2)
"1/e time"
T1/e = RC
(k = e)
(4)
In this experiment your instructor will assign each student a different value of k.
Data
1.
Chose one of the capacitors given and connect the circuit shown below.
© KFUPM – PHYSICS
revised 26/06/2010
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Department of Physics
Dhahran 31261
26-Jun-10
PHYS102 – 12
Note: IN THIS EXPERIMENT WE WILL BE USING POLARIZED CAPACITORS,
THAT IS, CAPACITORS WITH A POSITIVE AND NEGATIVE TERMINAL. IT
IS VERY IMPORTANT THAT CARE BE TAKEN TO HOOK THESE UP AS IN
THE DIAGRAMS, OTHERWISE THEY MAY BE DAMAGED. IF YOU ARE
UNSURE HOW TO DO THIS CONSULT THE INSTRUCTOR.
switch
S
47Ω
a
voltmeter
Power
supply
+
_
C
+
_
15.0
x 103
Ω
15 V range
b
2.
With the switch closed, turn on the power supply and adjust it until the voltmeter
reads some convenient voltage, say 10 volts. This is Vo .
3.
Open the switch and start the stopwatch. Measure the time for the voltage to fall from
Vo to Vo/k. This is T1/k.
4.
Use equation (3) to derive a relation between T1/k and RC, for your assigned value of
k, similar to equations (4).
Use the derived equation and the value R = 15.0 x 103 ohms to calculate the
capacitance of the first capacitor. (R is the effective resistance of the voltmeter.)
a
a
5.
Repeat with the second capacitor.
+
_
6.
Repeat with both capacitors in series.
7.
Repeat with both capacitors in parallel.
+
_
+
_
+
_
b
b
SERIES
PARALLEL
Optional
V vs t for a Discharging RC Circuit
Connect the resistor that you are provided in series with the voltmeter. In this case
discharging takes place much more slowly. Collect data to make a curve of V vs t in the
following way. Charge the capacitor to the full range of the voltmeter. Open the switch and
start your stopwatch. Record the reading of the voltage across the capacitor when t = 3 s.
© KFUPM – PHYSICS
revised 26/06/2010
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Department of Physics
Dhahran 31261
26-Jun-10
PHYS102 – 12
Charge the capacitor again to the full range of the voltmeter. Open the switch and start your
stop watch. Record the reading of the voltmeter at t = 6 s.
Record this process for t = 9 s, t = 12 s and higher values of t.
Plot curves of V vs. t and ln V vs. t. Draw a smooth curve through your point. The curve of
lnV vs. t should be a straight line. This can be proved by taking the logarithm of both sides of
eq. 3.
ln V(t) = ln Vo – ( t / RC )
Find the RC time constant from your graph. This is the time where V has fallen to 1/e of its
initial value. It is also equal to the negative inverse of the slope of the line in lnV vs. t.
Discussion and error analysis
1.
Calculate the expected series capacitance based on equation (1) and your
measurements in parts 4 and 5; what is the percent difference between this and the
measured value in part 6?
2.
Repeat the calculations in (1) for the parallel combination.
3.
What are the major sources of error in this experiment?
4.
Give a mathematical or physical argument why k cannot have values in the range
0 ≤ k ≤ 1. .
5.
Derive an expression for the current through the voltmeter while the capacitor is
discharging.
6.
What will be the final current through the voltmeter.
© KFUPM – PHYSICS
revised 26/06/2010
43
Department of Physics
Dhahran 31261